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Mind, Vol. 114 . 456 . October 2005

doi:10.1093/mind/fzi933

Kaplan 2005

Reading On Denoting on its CentenaryDavid Kaplan1

Part 1 sets out the logical/semantical background to On Denoting, including an ex-position of Russells views in Principles of Mathematics, the role and justification ofFreges notorious Axiom V, and speculation about how the search for a solution tothe Contradiction might have motivated a new treatment of denoting. Part 2 con-sists primarily of an extended analysis of Russells views on knowledge by acquaint-ance and knowledge by description, in which I try to show that the discom fiturebetween Russells semantical and epistemological commitments begins as far back as1903. I close with a non-Russellian critique of Russells views on how we are able tomake use of linguistic representations in thought and with the suggestion that a the-ory of comprehension is needed to supplement semantic theory.

My project is primarily expository and context setting. I also want tocorrect a few misunderstandings that Russell or I or others may havehad. Although I flag a few issues in my own voice, I am trying, on thewhole, to present my discussion and analysis in a way that is recogniza-bly Russellian. Part 1 sets out the logical/semantical background to OnDenoting, including an exposition of Russells views in Principles ofMathematics and speculation about how the search for a solution to thecontradiction Russell had discovered in Freges logic might have moti-

1 This paper is drawn from the course on On Denoting that I have taught at UCLA for morethan thirty years. I thought this was a good opportunity to produce class notes. A few months ago,when I read Alasdair Urquharts surprising Introduction to the invaluable, but very expensive,fourth volume ofThe Collected Papers of Bertrand Russellas well as Russells unpublished papersbetween Principles of Mathematics (1903) and On Denoting (1905), I came to better understandthat Russells attempts to avoid the contradiction he had found in Freges logic were related to hisworries about denoting. Urquharts discoveries made some old views about the relation betweenFreges logic and type theory relevant to Russells project. My discussion in Part 1 mixes historical

fact and speculation with logical fact and speculation. The unpublished papers before 1905 alsothrow light on Russells concerns about how we understand language. These concerns led to hisdistinction between knowledge by acquaintance and knowledge by description. My discussion ofthese issues, in Part 2, again mixes historical fact and speculation with, in this case, semantical andepistemological fact and speculation. I am not an historian, and though I have tried to read Russellcarefully, I am pitifully ignorant of the secondary literature. Even regarding the primary sources,Russell wrote more from 19001925 than I could read with adequate care in my lifetime. So thoughI write assertively, I expect scholars to find faults. I welcome correction. This paper is dedicated toVolume 4 ofThe Collected Papers of Bertrand Russell. It has benefited from the comments of JosephAlmog, C. Anthony Anderson, Benjamin Caplan, John Carriero, Timothy Doyle, Ruth Marcus,Donald A. Martin, Youichi Matsusaka, and Stephen Neale. Anderson caught one of the embarrass-ing errors in time for me to fix it. Martin provided useful discussion of Axiom V.

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Mind, Vol. 114 . 456 . October 2005 Kaplan 2005

vated a new treatment of denoting. I try to locate Russells earlier viewsin relation to Freges, and to frame the new treatment of denoting inopposition to these earlier views. In Part 2, I begin my examination ofOn Denoting itself, the logical, semantical, and epistemological thesesit proposes. I start with an analysis of Russells use of denoting phraseand follow with an extended discussion of Russells views on knowledgeby acquaintance and knowledge by description. I try to show that thediscomfiture between Russells semantical and epistemological com-mitments begins as far back as 1903. Part 2 completes my review of thefirst two paragraphs of On Denoting. I hope to write a sequel.

Part 1: Background

1.1 Principles of Mathematics

1.1.1 Language as a system of representation

It is, or should be, generally accepted that On Denoting (hereafterOD) is written in opposition to Russells own views in the chapter enti-tled Denoting in Principles of Mathematics (hereafter PoM).Thatchapter presupposes Russells view that language is a system for repre-senting things and arrangements of things in the world. The simple ele-ments of language stand for things and properties, and linguisticallycomplex expressions stand for complexes of those things and proper-ties. Russell calls the kind of thing that a sentence, the most importantlinguistically complex expression, stands for (or expresses, or means) aproposition. Hence, the constituents of propositions are the very thingsthat the propositions are about. For example, the sentence I met Bertieexpresses a proposition whose constituents are me, Bertie, and the rela-tional propertymeeting. All three of these constituents are entities to befound in the empirical world, according to Russell.2 A proposition (andany sentence that expresses it) is true if the way the things are arrangedin the world corresponds to the way the things are arranged in theproposition, in the case in question, if the relation ofmeetingactually

held between me and Bertie.Propositions have a structure, a kind of syntax of their own. Russell

often talks as if this syntax mirrored the syntax of natural language. InOD he modifies that view, as we shall see. The propositions exhibit allthe ways that the objects and properties of the world can be combinedin accord with this propositional syntax. What propositions there are is

2 Of course, if the sentence is about numbers or other non-worldly entities, the propositionalconstituents will not be worldly, but they will still be the things the proposition is about.



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determined by what objects and properties there are in the world. Whatsentences there are is determined by a narrower range of facts, includ-ing, for example, which objects and properties are of interest to the cre-ators of the language. Not every proposition need be expressed by asentence in an actual language. For Russell, his contemporaries, andthose that preceded them, it is the realm of propositions, existing inde-pendently of language, that form the subject matter of logic.3 One con-sequence of this propositions-before-language point of view is that thesymbolism used in the language of logic must be developed with greatcare. Our ability to study the logical relations among propositions may

be helped or hindered by how well the syntax of the language of logicarticulates with the structure of the propositions that form its subjectmatter.4

The view that language is a system of representation for the thingsand states (and possible states) of the world seems natural and appeal-ing, but it is not the only way to view language. Gottlob Frege, the greatcreator of modern symbolic logic and founder of Logicism, saw lan-guage as based on thought.5 On Freges picture, language is an externali-zation of, and thus a system for representing, thought. Freges meanings,unlike Russells, are elements of cognition and complexes of such ele-

3 Others may have had a somewhat different conception of the nature of propositions, but theview that the objects of logical study are prior to language was very widespread.

4 Russells conception seems to pose a challenge to the interpretation of modality in so far as it isconsidered possible for there seems to be things other than there are. This is because his proposi-tional functions seem not to be intensional in their domain. It may be possible to repair this di ffi-culty without either adding merely possible objects to the domains of propositional functions(which would have caused Russell to shudder) or replacing propositional functions with relationsamong properties (which would be unfaithful to the notion of a propositional function) by analyz-ing his notion of a function. Russells scepticism about modality, expressed in his unpublished 1905paper Necessity and Possibility (which can be found in The Collected Papers of Bertrand Russell,Volume 4 (hereafter CP4)) may have prevented him from ever confronting this challenge directly.

5 Logicism is the view that mathematical constants can be defined in pure logic, and that suchdefinitions provide a reduction of all truths of mathematics to truths of pure logic. Historically, itseems to have been part of Russells view that the truths of mathematics would be reduced truthsof logic that would provable in a single all-encompassing systematization of logic. For example, onp. 4 ofPoM, Russell describes the Kantian view, which asserted that mathematical reasoning is not

strictly formal. He then writes, with an excess of confidence, Thanks to the progress of symboliclogic, especially as treated by Professor Peano, this part of the Kantian philosophy is now capableof a final and irrevocable reputation. By the help of10 principles of deduction and 10 otherpremises of the general logical nature (e.g. implication is a relation), all mathematics can bestrictly and formerly deduced; and all the entities that occur in mathematics can be defined interms of those that occur in the above 20 premises. Frege is more cautious. Although he railsagainst loose standards of proof, he typically demands only the proof of the the fundamentalpropositions of arithmetic. For example, in his Grundlagen der Arithmetik, he writes on page 4e, we are led to formulate the same demand as that which had arisen independently in the sphereof mathematics, namely that the fundamental propositions of arithmetic should be proved, if inany way possible, with the utmost rigor; for only if every gap in the chain of deductions is elimi-



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ments.6 Like Russells propositions, Freges thoughts precede language.Frege claims that there is a repertoire ofthoughts common to all man-kind, and thus independent of the particulars of actual languages.

Frege used the word Sinne (senses) for the cognitive elements andcomplexes that are represented by linguistic elements and complexes.Thus, for Frege, the sense of a linguistic expression is what the expres-sion represents or means.7

Russell spoke of thought as something psychological, and stated thathis interest was in the objectof thought (now sometimes referred to asthe contentof a thought). Russell assumed, essentially without argu-

ment, that the kind of thing that served his semantic theory, i.e., histheory of meaning for language, was also the kind of thing that servedas an object of thought. So Russell also referred to the objects ofthought as propositions, and sometimes, perhaps to emphasize the factthat the constituents of propositions are the very objects that the prop-ositions are about, as objective propositions. Propositions are thus acommon element connecting linguistic representation with thought,and this provides a foundation for explaining our understandingof lan-guage. Although sentence meanings and objects of thought are thesame kind in Russell, he was aware that it did not follow that any prop-osition that could be represented linguistically could be an object ofthought.8

Frege does not distinguish thoughts from the objects or contents ofthoughts, as Russell did. Freges thoughts are also objective, but in a dif-

6 I speak a bit loosely here. Complex is Russells word. It is not clear that Freges meanings ac-tually have constituents in Russells sense. What seems common to the two views is that the mean-ings of complex expressions can be parsed into sub-meanings in a way roughly corresponding tothe way in which the complex linguistic expressions can be parsed into sub-expressions, thoughthe parsing of meanings might not exactly correspond to what a grammarian would tell us about

the parsing of expressions. (Frege remarks that active and passive constructions may have the samemeaning, and in Function and Concept he claims that (x)(x24x= x(x4)) has the samemeaning as x(x24x) = x(x4) (where the notation is for the course-of-values of a functionas discussed below). These examples suggest that Freges parsings may not be unique, and thusthat Freges meanings may not have a constituent structure. This I owe to Terry Parsons.

7 The reader will have noted that my use of italics goes one better than Russell, mixing togetherin one notation, reference to expressions and to their meanings along with the traditional use foremphasis.

8 It also doesnt follow that any object of thought could be represented through language, butRussell doesnt seem to have been interested in this.

nated with the greatest care can we say with certainty upon what primitive truths the proof de-pends In any case, we now know that not all truths of mathematics can be proved in a singleall-encompassing mathematical system of the kind envisaged, so Russells requirement is toostrong and should be replaced by the requirement to reduce truths to truths and proofs to proofs.



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ferent sense. The same thoughtcan be shared. So Freges thoughts arenot psychological in the sense of being subjective and unshareable.But they are certainly not objective in Russells sense of having worldlyobjects as constituents. This seems to leave Freges thoughts high anddry, divorced from reality (as thought can so easily be). So Frege postu-lates a second kind of representation whereby the elements and com-plexes of cognition represent worldly things. This second kind ofrepresentation, which he calls Bedeutung, is dependent on worldlyfacts; it is not determined by thought alone. The same elements ofthought, in other circumstances, could represent different objects, and

so the same thought could represent a different structure of worldly ele-ments. For Frege, it is through this second kind of representation thatthoughts, and ultimately sentences, come to be true and false.9

It is often said that the cognitions that Frege associates with a nameare in fact definite description-like in structure.10 This would explainthe relation of Bedeutung that holds between such a cognition and aworldly individual. But the explanation only works on the basis of aprior explanation of the Bedeutung relation that holds between thepredicates of the description-like cognition and (roughly) the classesof individuals to which they apply. This relation, which is left fairlymysterious, seems to be based on an implicit link (perhaps, identity)

9 Russell and Frege belonged to a mutual admiration society. There was, however, much mis-communication between them. It is my belief that a prime reason for this miscommunication wasthat neither ever quite understood or accepted that the others treatment of language was so fun-damentally different from his own. Russell says in PoMthat Freges semantical system is very muchlike his own. And he repeats this claim frequently. Neither ever seemed to fully grasp their funda-mental divergence over whether language is a system for representing things and states of theworld or things and states of the mind. This miscommunication was also engendered by the factthat they use the same language to mean very different things. In their correspondence, much ofwhich is published, one sees them frequently talking past one another with Frege trying to lay outhis conceptual apparatus in careful and precise detail, and Russell responding in terms of his ownconceptual apparatus, but using pretty much the same language. The difference between Russellspropositions and Freges thoughts lies at the heart of the difference between them. But these twonotions seem to have been conflated, perhaps because thoughts are for Frege, just as propositionsare for Russell, expressed by sentences and the objects of mental activity. It was also a very greatmisfortune that Frege had chosen to use the word Bedeutung for a notion close to what Russellcalled denotation. Russell translated Bedeutung in the customary way as meaning, which he con-trasted with denotation. What Russell meant by the English word meaning was much closer towhat Frege meant by Sinn, which Frege contrasted with Bedeutung. They corresponded in Ger-man, and, as far as I can tell, the translation problem never quite sunk in. Russell must have beenstupefied to see Frege write, as he did on December 28, 1902, You could not bring yourself to be-lieve that the truth-value is the meaning of a proposition. To which Russell responds on December12, 1904, [F]or me, the meaningof a proposition is not the true, but a certain complex which (inthe given case) is true. This disagreement is surely a problem engendered primarily by the transla-tion of Bedeutung. The correspondence is published in Freges Philosophical and MathematicalCorrespondence .

10 Im not sure that this is invariably correct, but it is often said.



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between the elements of cognition that are represented by wordslike red, hot, dog, and star, and the properties and relations thatRussell considered worldly. We dont have presentations of theclasses themselves, so it seems that we must rely on presentations ofproperties (which, given the actual facts, could determine the classes).The simplest hypothesis is that the cognitions in question are the prop-erties and relations. An alternative hypothesis is that the cognitions inquestion come to be of(in a third wayof representation) such proper-ties and relations through presentations thereof. A difficulty with thefirst is that the cognitions themselves are supposed to be innate.11 If

Frege were to abandon the innateness claim, and accept Russells viewthat we become acquainted with worldly property and relationsthrough experience, he could bring properties and relations directlyinto the realm of thought. But he would then face the worry that thesame property might be presented in ways that we fail to identify, forexample, we may fail to recognize every presentation of the property ofbeing a dog(Dogs range in size and form from the diminutive Chihua-hua to the monstrous Great Dane, and every size and shape imaginablein-between making the domestic dog, Canis familiaris, the most variedspecies on the face of the planet.), and distinct properties, for example,the property of being a planetand the property of being a star, might bepresented in ways that led us, mistakenly, to identify them. These errorscould cause us to mistake one thought for another. This would be aserious difficulty for Freges theory, since it was designed to explainerrors of recognition from a standpoint that was free of them. We willreturn to worries about recognition.

If we put these concerns aside for now (a big IF), and suppose thatthe elements of cognition represented by words like red, hot, dog,and star are Russellian properties (or if we close the gap betweenFregean senses of these words and Russellian properties in some otherway), Fregean thoughts become worldly and a subcategory of Russellspropositions. Bedeutung can then be thought of as playing two roles.First, it assigns to a Russellian property the function which assigns

Truth to every individual that has the property and Falsehood to everyindividual that lacks it. This assignment is a factual, empirical matter,since the property alone does not determine the individuals of which itholds.12 Second, Bedeutung calculates the values of all complexes, basi-

11 This difficulty might be avoided by adopting a very strong form of rationalism, of the kindsometimes advocated by Chomsky.

12 At least not on Russells metaphysics of the time, which allowed individuals to be simples. Evenif individuals were bundles of properties, the property alone would not know which bundles existed.



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cally by applying functions to arguments, including applying higherorder functions to first order functions in ways that neednt concern ushere. The calculational role is not empirical, because all the informa-tion required for the calculation is contained in the functions on whichthe calculation is performed. The result of this second role of Bedeu-tung, the calculational role, is that each definite description will calcu-late out to an individual, and each sentence will calculate out to truthvalue.13

Fregean thoughts are not, except in exotic cases, about their constitu-ents, they are about the Bedeutung of their constituents. It is odd to say

that a sentence is about its truth value, but natural to say that Theauthor ofWaverleywrote Guy Mannering is about the author ofWaver-ley, namely, Sir Walter Scott.

Returning now to Russell, we can see how his view of the representa-tional role of language leads directly to the claim that sentences areabout what their elements represent (think of I met Bertie) and prop-ositions are about their own constituents. So Russell seems to have noneed for additional theoretical resources to describe what a propositionis about. However, I see no reason why Russell could not introduceBedeutung (in both of its roles) explicitly into his semantics. It mustappear, at least implicitly, in any calculation of whether a proposition istrue and of what a denoting concept(see below) denotes.

Like Bedeutung, truth is an empiricalproperty of propositions. Nowit is natural to try to stay away from the grossly empirical in a semantictheory. Our theory of language should capture features of syntax andsemantics that explain the use of language by competent speakers. Itneednt tell us which sentences are true. Thats a job for the special sci-ences. On the other hand, although semantics need not tell us whichsentences are true, it should explain what it is for a sentence to be true.And for this latter task, the notion of Bedeutungis useful. Perhaps wemay conclude that semantics should tell us which proposition (orthought) a given sentence represents (perhaps in terms ofwhich con-stituents the constituents of the sentence represent) and should

explain what it is for a proposition (or thought) to be true (perhaps interms of what it is for the constituents to have a particular Bedeu-

13 This is a description of Fregean semantics given from a Russellian perspective, one which, forexample, takes the notion of worldly properties and relations for granted. If one could take forgranted the first Bedeutung relation, the empirical one that holds between the senses of predicateslike _ is red, _ is hot, _ is a dog, and _ is a star and the functions which assigns Truth to everyindividual that satisfies the predicate and Falsehood to every individual that does not (the very re-lation that I called fairly mysterious in the preceding paragraph), Fregean semantics might lookmuch more uniform and elegant. However, there would still, I believe, be the two quite differentroles for Bedeutung to play.



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tung).14 Understanding what it is for a proposition (or thought) to betrue is part of our understanding of what language can be used to do.

Although Russell presents his semantics as Bedeutung-free, thenotion does rear its head in one quasi-epistemological corner of Rus-sells theory, denoting.

1.1.2 Denoting phrases in Principles of MathematicsThe Denoting chapter (Ch.5) lays out an exception to the principlethat propositions are about their constituents. In the case of certaincomplex linguistic phrases, in particular but not exclusively, thoseformed with the six determiners all, every, any, a, some, and the,

the corresponding constituent of the proposition is itself to be a com-plex. But the proposition is not about this complex; it is instead aboutwhat the complex denotes, an object that is usuallynota constituent ofthe proposition, and often not even known to the speaker. I have givenone example, here is another: The proposition expressed by George IVembarrassed the author ofWaverley may be about George IV, embar-rassing, the novel Waverley, and authorship, but it is also about Sir Wal-ter Scott, who is the author ofWaverleyand the man whom George IVis said to have embarrassed. Scott does not appear to be a constituent ofthe proposition, and the reporter may not even have known that theman George IV embarrassed was Scott, still the proposition is, in part,

about Scott.Linguists call these phrases determiner phrases because of their syn-

tactical structure; they are constructed from determiners. Russell calledthem denoting phrases because of their semantical property; they arephrases that denote.15 Russell called the complexes they express denot-ing complexes or sometimes denoting concepts because they are com-plexes (or concepts) that denote.16 Denoting complex better conveyswhat Russell had in mind, but in PoMhe uniformly used denotingconcept, so we will follow him in that usage; it is a distinction without adifference. A proposition containing a denoting concept is not about

14 How our semantics tells us which constituent of a Russellian proposition a name represents is

a delicate matter. We dont want our semantics to resolve the truth of all identities between names.Russells own solution, which uses definite descriptions and mixes semantics and epistemology isnot ultimately satisfactory.

15 It is plain, to begin with, that a phrase containing one of the above six words always denotes.PoMsect. 58.

16 In OD, by which time their existence had become dubious, Russell uses denoting complexes.Calling them concepts was not, for Russell, a covert way of making them more mentalistic. Con-cept was used more in the sense of a classifier. In PoM, all properties and relations are regarded asconcepts (though not, of course, as denoting concepts). Frege also used concept (Begriff) in a clas-sificatory, completely non-mentalistic way.



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the concept but about what the denoting concept denotes. As Russellmight have put it, George IV didnt embarrass a denoting concept, howwould he do that; he embarrassed the denotation of the denoting con-cept, namely, Scott. Though both the linguistic phrase and the conceptit expresses are said to denote (and though they presumably denote thesame thing), in this chapter ofPoMRussell seems to focus primarily onthe denoting of the propositional constituent, the denoting concept,though it is hard to tell because of Russells characteristic indifferenceto the distinction between linguistic expressions and what theyexpress.17

[T]he fact that description is possiblethat we are able, by the employmentof concepts, to designate a thing which is not a conceptis due to a logicalrelation between some concepts and some terms [for Russell, term is prob-ably best read as individualor possiblyentity],18 in virtue of which such con-cepts inherently and logicallydenote such terms [individuals]. It is this senseof denoting which is here in question. A concept denotes when, if it occursin a proposition, the proposition is not aboutthe concept, but about a term[individual] connected in a certain peculiar way with the concept. If I say Imet a man, the proposition is not about [the denoting concept] a man: thisis a concept which does not walk the streets, but lives in the shadowy limboof the logic-books. What I met was a thing, not a concept, an actual man witha tailor and a bank-account or a public-house and a drunken wife. If we

wish to speak of the concept, we have to indicate the fact by italics or invertedcommas.19, 20, 21

Denotingconcepts are anomalies, exceptions to the rule and difficult toexplain. Yet Russell attached great importance to their role and to thedenotingrelation.

17 In OD, when denoting concepts have been banished, he returns to denoting phrases, and pro-vides an exceedingly thin sense of denoting that applies only to proper de finite descriptions(namely, those that succeed in describing exactly one thing).

18 In PoMsection 47, Russell writes, Whatever may be an object of thought, or may occur inany true or false proposition, or can be counted as one, I call a term. This, then, is the widest wordin the philosophical vocabulary. I shall use as synonymous with it the words unit, individual, andentity. In later developments, he gives slightly conflicting explanations (as readers of Russellwould expect).

19 PoMsect. 56. Here, as in all subsequent quotations, bracketed insertions are my comments.

20 In British English, inverted commas is simply synonymous with what Americans call quo-tation marks. I have seen reprints of OD in which American editors seem to have struggled to finda special notation for inverted commas, especially within the notorious Grays Elegy passage. In theoriginal, Russell invariably uses double quotation marks except for quotation marks within quota-tion marks.

21 Russell here leaves the false impression that the concept a man denotes the actual man hemet. As we shall see, he explicitly rejects this view. His stereotyping of social classes may also leavea false impression of his views. I am less certain of this.



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This notion [denoting] lies at the bottom (I think) of all theories of sub-stance, of the subjectpredicate logic, and of the opposition between thingsand ideas, discursive thought and immediate perception. These various de-velopments, in the main, appear to me mistaken, while the fundamental factitself, out of which they have grown, is hardly ever discussed in its logical pu-rity.22

Given all this to denotings credit, it seems like a lot to sweep away; yetthe purpose of OD is, I believe, to sweep awaydenoting. But first, let uslook at what progress Russell felt he had made in PoMin the analysis ofthe denoting of his six kinds of denoting phrases. He carefully studied

denoting phrases that used all, every, any, a, and some, distinguish-ing subtle differences in shades of meaning.23, 24 He then sets out anapparatus to account for denoting by introducing a new kind of object,conjunctions and disjunctions of individuals (he calls them combina-tions of terms), which will serve as the denotation of certain denotingphrases.

The combination of concepts as such to form new concepts, of greater com-plexity than their constituents, is a subject upon which writers on logic havesaid many things. But the combination of terms [individuals] as such, toform what by analogy may be called complex terms [complex individuals],is a subject upon which logicians, old and new, give us only the scantiest dis-cussion. Nevertheless, the subject is of vital importance to the philosophy of

mathematics, since the nature both of number and of the variable turnsupon just this point.25

He first explains his idea in terms, not of determiner phrases, but sen-tences with complex grammatical subjects like Brown and Jones arecourting Miss Smith and Miss Smith will marry Brown or Jones. Heclaims that in such contexts, the complex expressions Brown andJones and Brown or Jones each denote a certain combination of theindividuals Brown and Jones: in the first case, a kind of conjunction ofthem, and in the second case, a kind of disjunction of them. Of the

22 PoMsect. 56.

23 Interestingly, the, which is to figure so centrally in OD is given relatively short shrift in PoM

(it is discussed in connection with definitions and identity sentences). It is the white sheep of thestory; the problem of how to deal with improper definite descriptions (those that do not describeexactly one thing) is not even mentioned.

24 In some cases, Russell seems to be attempting, in his analysis of the denotation of a denotingphrase, to accomplish what in OD he (and modern logicians) would accomplish through the no-tion of scope. For example, he argues that a point (as contrasted with some point) denotes avariable disjunction of points because, a point lies between any point and any other point; butit would not be true of any one particular point that it lay between any point and any other point,since there would be many pairs of points between which it did not lie. [PoMsect. 60.]

25 PoMsect. 58.



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that a word can be framed with a wider meaning thanterm raises gravelogical problems.28, 29

When he concludes that the five determiner phrases (excluding definitedescriptions) all men, every man, any man, a man, and some mandenote distinct objects, he worries about the objects his theory postu-lates.

It appears from the above discussion that, whether there are different waysof denoting or not, the objects denoted byall men, every man,etc. are cer-tainly distinct. It seems therefore legitimate to saythat the whole differencelies in the objects, and that denoting itself is the same in all cases. There

are, however, many difficult problems connected with the subject, espe-cially as regards the nature of the objects denoted. Consider again the

proposition I met a man. It is quite certain, and is implied by this

proposition, that what I met was an unambiguous perfectly definite

man: in the technical language which is here adopted, the proposition is

expressed by I met some man. But the actual man whom I met forms

no part of the proposition in question, and is not specially denoted by

some man. Thus the concrete event which happened is not asserted in theproposition. What is asserted is merely that some one, of a class of con-

crete events took place. The whole human race is involved in my asser-

tion: if any man who ever existed or will exist had not existed or been

going to exist, the purport of my proposition would have been different.

Or, to put the same point in more intensional language, if I substitute

for man any of the other class-concepts applicable to the individualwhom I had the honour to meet [for example, student], mypropositionis changed, although the individual in question is just as much denoted

as before [i.e. there is just as much reason to think that the actual man is

denoted]. What this proves is, that some man must not be regarded as ac-tually denoting Smith and actually denoting Brown, and so on: the

whole procession of human beings throughout the ages is always rele-

vant to every proposition in which some man occurs, and what is denotedis essentially not each separate man, but a kind of combination of all

men [presumably, the constant disjunction of all men discussed

above].30, 31

28 PoMsect. 58.29 This is one of my favourite places where Russell carefully notes what may be an insuperable

difficulty, and then continues to move ahead. Russell took an admirably experimental attitude to-ward philosophical theories.

30 PoMsect. 62.

31 I dont see how changing the proposition by replacing the denoting concept some man bysome student(assuming the man also to be a student) helps to show that the denoting conceptsdont denote the actual man. Changing the man himself would show it, if we added the tacit as-sumption that some man has the same denotation in each of its uses.



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Russell concludes this discussion with a rather sceptical reflection.

There is, then, a definite something, different in each of the five cases,

which must, in a sense, be an object, but is characterized as a set of terms

[individuals] combined in a certain way, which something is denoted byall men, every man, any man, a man or some man; and it is with this veryparadoxical object that propositions are concerned in which the corre-

sponding concept is used as denoting. [Underlining added.]32

The tentativeness of Russells views about denoting and the theory ofpropositional functions and variables that he built upon them are quiteexplicit. In the chapter on propositional functions, he writes,

The subject is full of difficulties, and the doctrines I intend to advocate areput forward with very little confidence in their truth.33

In the chapter on the variable, he writes,

Thus in addition to propositional functions, the notions ofanyand of denot-ing are presupposed in the notion of the variable. This theory, which, I ad-mit, is full of difficulties, is the least objectionable that I have been able toimagine.34

Worries about what he usually called the Contradiction hover in thebackground ofPoM, and are sometimes addressed directly.35 But theyare not the main object of the book.

The present work has two main objects. One of these, the proof that all pure

mathematics deals exclusively with concepts definable in terms of a very smallnumber of fundamental logical concepts, and that all its propositions are de-ducible from a very small number of fundamental logical principles, is under-taken in Parts II.VII. of this Volume, and will be established by strict symbolicreasoning in Volume II. The other object of this work, which occupies PartI., is the explanation of the fundamental concepts which mathematics acceptsas indefinable. This is a purely philosophical task, and I cannot flatter myselfthat I have done more than indicate a vast field of inquiry, and give a sampleof the methods by which the inquiry may be conducted.36

32 PoMsect. 62. When talking about propositions, Russell seemed to use is concerned with andabout synonymously. There is a rather clear example at the end of his Descriptions chapter inIntroduction to Mathematical Philosophy, hereafter IMP.

33 PoMsect. 80.34 PoMsect. 86.

35 In the Preface to PoM, Russell writes, In the case of classes, I must confess, I have failed toconceive any concept fulfilling the conditions requisite for the notion class. And the contradictiondiscussed in chapter ten [titled The Contradiction] proves that something is amiss, but what thisis I have hitherto failed to discover. In Appendix B, which adumbrates the theory of types, hewrites on the last page of the book of a closely analogous contradiction [concerning the totality ofall propositions] which is probably not solvable by this doctrine.

36 PoMPreface to the first edition.



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There can be no doubt that On Denoting is a direct attack on thedenoting concepts of Chapter V ofPoMand on the very paradoxicalobjects they were said to denote. It is the central tenet of OD thatdenoting phrases have no meaning in isolation, which is the OD wayof saying that there is no propositional constituent corresponding to adenoting phrase (at least none that corresponds in the way that propo-sitional constituents correspond to names, nouns, and adjectives). Hereis the very different view of OD:

Everything, nothing, and something, are not assumed to have any meaningin isolation, but a meaning is assigned to everyproposition [sentence]

in which they occur. This is the principle of the theory of denoting I[now] wish to advocate: that denoting phrases never have any mean-ing in themselves, but that every proposition in whose verbal expres-sion they occur has a meaning. The difficulties concerning denotingare, I believe, all the result of a wrong analysis [such as that given inChapter V ofPoM] of propositions whose verbal expressions containdenoting phrases.37,38

1.1.3 Why did Russell abandon denoting concepts?

I had always assumed that the reason for Russells change of heartregarding denoting concepts was the difficulty of making the PoMthe-ory work, especially for such denoting concepts as some man, whosedenotation was to be one of those very paradoxical objects, the disjunc-tion of all the men.39 The alternative was Freges elegant theory of quan-tifier phrasesessentially Russells everything, something, andnothingas higher order functions on first order functions from indi-viduals to truth values.40 Frege treats scope as scope. Russell reports thatFreges theory was not known to him when he was writing PoM.

Professor Freges work, which largely anticipates my own, was for the mostpart unknown to me when the printing of the present work began; I had seenhis Grundgesetze der Arithmetik, but, owing to the great difficulty of his sym-bolism, I had failed to grasp its importance or to understand its contents.

37 OD p. 480.

38

Since for Russell, the proposition expressed is the meaning of a sentence, the second clause ofthe principle might be rephrased tautologically as every meaningful sentence in which a denotingphrase occurs has a meaning. Russell's regular use of proposition for both sentence and meaningof a sentence requires vigilance, but only rarely leads him astray.

39 I am not suggesting that it could not be made to work. Quite the contrary, I think it, or some-thing approximating it, could be made to work. For an example, see Parsons ( 1988). Even the ac-counting for scope in terms of the object denoted might be made to work, provided we canaccount for the object denoted in terms of the scope of the denoting phrase, as Russell sometimesseems to do.

40Almost, but not quite, Russellian propositionalfunctions.



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The only method, at so late a stage, of doing justice to his work, was to devotean Appendix to it If I had become acquainted sooner with the work ofProfessor Frege, I should have owed a great deal to him, but as it is I arrivedindependently at many results which he had already established. 41

Russell had already been careening toward Freges understanding ofquantification in his PoMtreatment of what he called formal implica-tion in the language of logic and mathematics; for this he offered asemantic theory in terms of variables and propositional functions.However, in accordance with the theory of denoting in Chapter V, heargued that the formal implication ifx is a man then x is mortal

(understood as saying that the corresponding propositional function istrue for all values of the variable) expressed a proposition that was dis-tinct from, though equivalent to, that expressed by every man is mor-tal.

consider the proposition [sentence] anya is a b. This is to be interpret-ed as meaning [i.e. translated into the language of logic and mathematics as]xis an a implies xis a b. [This is Russells standard formulation of the for-mal implication, understood as holding for all values of the variable x.] Itis plain that, to begin with, the two propositions [sentences] do not mean thesame thing: for any a is a [denoting] concept denoting onlyas, whereas inthe formal implication xneed not be an a. But we might, in Mathematics,dispense altogether with anya is ab, and content ourselves with the formal

implication: this is, in fact, symbolically the best course.42

In sum, the view ofPoMseems to be that there are two languages, thenatural language, which contains denoting phrases, and the much moreconstrained language of logic and mathematics, which contains openformulas and formal quantifiers. The semantic theory for the formerwould involve denoting concepts; but the semantic theory for the lattercan make do with more limited means, perhaps just propositional func-tions and their properties. Many sentences in the denoting phrase lan-guage can be translated into sentences of the formal quantifierlanguage. The propositions expressed by such a sentence and its trans-lation will be logically equivalent, but distinct. The grammar of natural

language sentences was taken as a guide to the structure of the proposi-tions expressed.

41 PoMPreface to the first edition. Notice the graciously confident counterfactual in the finalsentence.

42 PoMsect. 89. I realize that every man is mortal isnt quite of the form any a is a b. Russellsobsession with the determiner any is a story I do not fully grasp and have no desire to tell.



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On the whole, grammar seems to me to bring us much nearer to a correctlogic than the current opinions of philosophers; and in what follows, gram-mar, though not our master, will yet be taken as our guide.43

I had assumed that the abandonment of denoting concepts in OD refl-ected the fact that when Russell became better acquainted with Fregestheory, he threw in the towel on denoting concepts, and simply used hisown variant of Freges superior theory of quantification.44 In OD hissemantics can be read as if he had tacitly translated the denoting phraselanguage into the language of logic and mathematics, and then givenhis semantical analysis for the sentences in thatlanguage. What Russell

claimed to be symbolically the best course for the language of logicand mathematics (by which I assume he meant the best symbolism forlogic and mathematics) is seen in OD as the best understandingof thedenoting phrase language. This has the consequence that where in PoMwe had equivalent but distinct propositions, we now have a single prop-osition. Translating a sentence of the denoting phrase language into asentence of the language of logic is no longer seen as yielding a distinct(but equivalent) proposition, but rather as revealing the pre-existing,but hidden, logical form of the denoting phrase sentence.

One of the consequences of the shift is that the burden of establishingthe equivalence of the sentences of the two languages moves from thescience of logic to the art of translation (or symbolization as it is nowcalled). This is the affliction that Russell bequeathed to our logic stu-dents.

One thing puzzled me. The one denoting phrase whose denotationdid notseem to require the postulation of a very paradoxical object isthe definite description, given short shrift by Russell in PoM(thoughmade central by Frege).45 So why devote 80% of OD to redoing the the-ory of definite descriptions?46 The worries about very paradoxicalobjects in Russells PoMtheory of denoting may be good reason to

43 PoMsect. 46.

44 Russells variant involves propositional functions rather than Freges truth valued functions.

45 As noted, Russell did not so much as mention the possibility of a definite description beingimproper in PoM, whereas for Frege, the definite description, with its two kinds of meaning, be-came the paradigm of a meaningful expression.

46 Russell does argue in OD that Freges sense and denotation theory of descriptions fails to givetruth values to certain sentences that should, intuitively, have them. But these arguments seemmore of a justificatory afterthought than the real motivation for his drive to rid logic of such ex-pressions. This part of Freges theoryfirst appears explicitly in ber Sinn und Bedeutung; hereaf-ter S&B. The main ideas are anticipated at the end of section 8 of Freges Begriffsschrift(1879).Freges article is translated and reprinted almost everywhere that OD appears. Beware versions ofS&B in which the title is translated as Sense and Meaning, lest you fall into Russells misunder-standings of Frege.



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favour Freges treatment of the quantifierdeterminer phrases all men,every man, any man, a man, and some man, but those reasons didnot seem to argue for a similar recasting of the semantics of definitedescriptions. Indeed, Frege showed the way bynottreating them simi-larly, and Russell knew it.47 Furthermore, it couldnt be, as Strawson(1950) would insist, that Russell was motivated by a concern to find atreatment of definite descriptions that ensured that sentences contain-ing improper descriptions remained meaningful. Russells treatment ofdefinite descriptions in PoMalreadygave meaning, even a meaning inisolation, to all definite descriptions, proper as well as improper. So I

concluded that when Russell started eliminating denoting phrases (byimplicitly translating into the language of logic), he just got carriedaway.

I was wrong.

1.2 The Contradiction

1.2.1 Urquharts Discovery

In his illuminating Introduction to Russells papers in logic during theperiod 1903 to 1905,48 Alasdair Urquhart uses Russells correspondenceduring the period between PoMand OD to demonstrate that the goalof the development of the theory of descriptions in OD was to find a

way around the Contradiction. As Urquhart writes,Most of the very voluminous secondary literature on Russells Theory of De-scriptions discusses it in isolation from its setting in the enterprise of the log-ical derivation of mathematics; the resulting separation of the logical andmathematical aspects of denoting is foreign to Russells own approach.49

It is a simple historical fact that Russells work on denoting was done inthe course ofhis attempts to solve the contradiction. But we now knowthat Russell himself saw his work on denoting as in aid ofthat project.

Here is an eye-opening passage from an April 14, 1904 unpublishedletter unearthed by Urquhart:

Alfred [North Whitehead] and I had a happy hour yesterday, when we

thought the present King of France had solved the Contradiction; but itturned out finally that the royal intellect was not quite up to that standard.50

47 See the discussion of Frege in OD on p. 483.

48 Introduction to CP4; hereafter Introduction.

49 Introduction p. xxxii.

50 From a letter to Alys Pearsall Smith, Russells then wife, quoted in Introduction p. xxxiii.



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This unmistakably connects the problem of how to treat improperdefinite descriptions with the Contradiction.

In a previously published retrospective letter of March 151906, Rus-sell wrote,

In April 1904 I began working at the Contradiction again, and continued atit, with few intermissions, till January1905. I was throughout much occupiedby the question of Denoting, which I thought was probably relevant, as itproved to be. The first thing I discovered in 1904 was that the variable de-noting function is to be deduced from the variable propositional function,and is not to be taken as an indefinable. I tried to do without as an indefina-ble, but failed; my success later, in the article On Denoting, was the sourceof all my subsequent progress.51

What Russell discovered seems to be that first, a singular denotingphrase like xs father, which Russell may have thought of as standingfor a function from individuals to individuals (a denoting function), canbe put into a standard form by introducing the formula yfathered x,52

which may be thought of as standing for a propositionalfunction, andthen using the iota operator, which picks out the unique argument tothe propositional function that yields a true proposition. This allows usto form a definite description that can replace the original denotingphrase. So the singular term xs father can be put into the standardform of a definite description, the ysuch that yfathered x. This does

not rid us of singular denoting phrases, but at least we have put them allin one form.

Perhaps it was this consolidation and focus on uniqueness thatbrought Russell to the critical insight for the second step, which we mayput as follows: that definite descriptions are nothing more than indefin-ite descriptions with uniqueness added. Thus, the definite descriptionsthe ysuch that yfathered x can be transformed into the indefinitedescription a ysuch that (yfathered xand onlyyfathered x). Russellwas already translating sentences containing indefinite descriptionsinto a formalism using existential quantification. So the so-called con-textual elimination of definite descriptions can be seen as reducing to

two steps, first the replacement of the definite description by anindefinite description with uniqueness added, and second, the contex-tual elimination of sentences containing the indefinite descriptions infavour of what amounted to existential generalizations. The second step

51 Russell always insisted on writing this in biblical form as ybegat x.

52 From a letter to Philip Jourdain, quoted in Introduction p. xxxiii. But note that all my sub-sequent progress here refers to only a nine month period.



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is something Russell took for granted when working in the language oflogic.53,54

In this way, all singular denoting phrases, i.e., complex singularexpressions, are ultimately eliminated from the language.55 The inter-esting question at this point is why did Russell think this a worthy goal,and why did he think it a help in resolving, or avoiding, the Contradic-tion? Most authors think that it was the theory of types that avoided theContradiction and that the elimination of singular denoting phraseswas irrelevant.

It may be that part of the importance that Russell attached to his the-

ory of descriptions was really due to the liberating effect of what hecalled his principle of denoting: that although denoting phrases have nomeaning in isolation, we can systematically explain the meaning ofevery sentence in which they occur. Russell came to call expressions towhich this principle applies incomplete symbols.56 A significant use ofthis idea occurs in PMwhere the expression for the extension of a prop-ositional function is treated as an incomplete symbol.57 However, thetreatment ofextensions of propositional functions is fundamentally dif-ferent from that of definite descriptions. Whereas the theory of descrip-tions analyzes the use of definite descriptions in terms of whether or

53 This is not the way Russell usually puts it, though he comes close to this formulation in the

first paragraph of his discussion of the in OD. This very clever idea may suffice for Russells pur-poses, to found mathematics on logic, but probably does not work in general, for example in thecase of Some Greeks worshipped the sun-god. (This case is drawn from an example of AlonzoChurchs and from Russells analysis of Apollo as abbreviating a definite description.) Note thatthe problem does not lie in the treatment of definite descriptions as indefinite descriptions withuniqueness added (which does seem to work), but rather in the second step, the treatment of in-definite descriptions as interpretable by existential generalizations.

54 Urquhart reports in his Introduction that Peano had already suggested the device of contex-tual definition in a monograph on mathematical logic read by Russell. Although Peanos operatorselects the unique element of a class rather than the unique thing satisfying a description, the con-nection with Russells contextual definition of definite descriptions is obvious.

55 I recognize that I am speaking somewhat loosely about contextual eliminations. But theamount of apparatus required to be precise, especially about scope in natural language, is morethan the purposes of this article can carry. Russells efforts in this direction amount to his saying: I

use C(x) to mean a proposition in which xis a constituent and then telling us in a footnote thatC(x) really means a propositional function. It seems that C is for context, but the switch betweenlinguistic context, when he writes things like C(no men), and the propositional function ex-pressed by C(x), when he writes things l ike C(x) is always true, beclouds his exposition.

56 See Ch. 3, Incomplete Symbols, of the Introduction to Principia Mathematica; hereafter PM.

57 Propositional functions are intensionalin exactly the following sense: two propositional func-tions may assign true propositions to the same individuals while remaining distinct. The exten-sions of such propositional functions should be such that if distinct propositional functions assigntrue propositions to the same individuals the extensions of the two functions will be identical. Anatural way to think of the extension of a propositional function F is as a function that assigns to



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not there exists a unique individual so described, PManalyzes sen-tences containing expressions for the extension of a propositional func-tion in a way that makes it irrelevantwhether or not there exist suchthings. The PMtheory is a theory ofvirtualextensions; we have the sin-gular denoting phrases that purport to denote such things, but the talkof such things is explained away as talk of other kinds of things, so intu-itively, theyneverdenote.58 Thus the treatment of extensions in PMisnot based on the treatment of definite descriptions, that is, it does notreduce such expressions to definite descriptions. The treatment ofextensions is based rather on the principle that if an explanation can be

given for every sentential context in which a given expression appears,no further assignment of meaning to the expression itself isrequired.59, 60 One way of assigning meaning to every sentential contextis to start by assigning a meaning to the given expression, but Russellsdevelopment of his theory of descriptions showed him that there wereother ways.

I think it not unlikely that it is the liberating effect of his principle ofdenotingthe opportunity to sweep away troublesome entities infavour of the virtualthat Russell had in mind in attributing all mysubsequent progress on the Contradiction to his theory of descrip-tions. But I do not think the PMtreatment of extensions of proposi-tional functions is important from a logical point of v iew,61 so I

58 I say intuitively because we dont have a precise definition ofdenotingin the case of exten-sions in PMlike the one Russell gives us in OD for definite descriptions. Note that the definition inOD does correspond to the intuitive notion.

59 This method promised to be much more useful than I think it has turned out to be in thehands of those philosophers who have used it. It is a natural treatment for only a few singular ex-

pressions, such as the average sophomore in The average sophomore enrolls in 4.2 classes andcompletes 3.8 of them. One may hesitate to use Russells theory of descriptions in such a case sinceit requires the hypostatization of an abstract sophomore. It seems more natural to understand allcontexts of the average sophomore in terms of statistical claims about the actual sophomores.However, in my view, it is usually better to hypostatize, perhaps even in a case like this, and cer-tainly in the treatment of extensions of propositional functions.

60 Gdel (1944) expresses doubt that Russell has given a meaning to all such sentential contextsbecause the syntax ofPMis so ill-explained that it is impossible to tell what all the sentential con-texts are.

61 It may have been historically important to philosophers who sought to ape its method.

each individual xthe truth value of the proposition F(x). Instead of truth values, anyfixed pair ofa true and false proposition would suffice. The extension of a propositional function can also bethought of as the characteristic function of a class (the class of individuals to which the extensionof the function assigns Truth (or the fixed true proposition). Russell seems to have thought of ex-tensions of propositional functions this way, calling them classes. It is thus, in replacing extensionsof propositional functions with virtual extensions, that Russells No Class theory comes about.



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continue to look for more interesting ways in which the theory ofdescriptions might have been seen as relevant to the Contradiction.

In the following I will show how the problem regarding the Contra-diction in Freges Grundgesetzefirst appears as a problem about denot-ing. This would make it natural for Russell to think that a new theory ofdenoting could be helpful in clarifying that part of the problem.

It takes some time to tell the story, but it is an interesting and impor-tant story, worth the telling in its own right. However, much of it is tan-gential to OD. So the impatient reader can take my word for it, andjump directly to Part 2 on page 968.

1.2.2 The incompleteness of linguistic expressions and functionsThe language of mathematics is largely a language of operation symbols(i.e. functional expressions, like + and ) rather than predicates, andFreges logic was well-suited to it.62 The language freely allowed func-tions from entities of every kind to objects. Among the primitive signsof the language is an operator which, when applied to a functionalexpression, yields the name of the course-of-values of that function.63

In Freges metaphysics there is a fundamental divide between func-tions (which are incomplete or unsaturated things) and objects (whichare complete or saturated things). The distinction seems to derive fromFreges syntactic view that a sentence like Dion walks should be parsed

into components of which one contains the gap resulting from the lit-eralremoval of the other from the whole. Thus the parts may be Dionand _ walks (or perhaps walks and Dion _). In the case of true func-tional expressions, for example (2 + 3x2)x Frege was especially con-cerned to isolate the function name from the argument expression.

The essence of the function manifests itself in the connection it establishesbetween the numbers whose signs we put for x and the numbers that thenappear as denotations of our expression Accordingly the essence of thefunction lies in that part of the expression which is over and above the x.The expression for a function is in need of completion, unsaturated.64

62 I refer here to the ill-fated system ofGrundgesetze. The introductory sections, plus the Ap-pendix to Volume II in which Frege discusses the Contradiction, are translated by MontgomeryFurth as The Basic Laws of Arithmetic; hereafter Basic Laws. All translations are taken from BasicLaws.

63 Course-of-values is Furths translation in Basic Laws of Freges Werthverlauf . It is unrelated toso-called course-of-values induction. Freges term is also often translated as value range orrange-of-values. When the function is a concept (i.e. a function to the two truth values) it is alsoreferred to as the extension of a concept; Frege talks this way, but it is dangerous talk.

64 Section 1 ofGrundgesetze, emphasis in the original.



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We see the characteristic Fregean elegance in this coordination of lin-guistic incompleteness with the incompleteness of a semantically asso-ciated entity.

But is Freges argument for linguistic incompleteness plausible? Heseems to be proposing a syntactic theory according to which com-pounds are built from incomplete expressions. However, syntactictheory doesnt work like that. The syntactic operations (functions) thatyield compounds from components neednt do so byfilling gaps. Fur-thermore not every literal removal of a part yields a function that isimplicit in the parsing. So for example, in Bertie met the father of

Charles IV the function Bertie met the father of _ does not represent astage in the parsing of the former. Perhaps the syntactic functions thatyield compound expressions when applied to their parts are unsatu-rated (though I see no reason why they need be), but there is certainlyno need for incomplete expressions. Basically, nothing(that is, no well-formed part of language) remains when we remove Charles IV fromBertie met the father of Charles IV, nothing more than what remainswhen we remove the cat from cattle (to cite a well-known example ofQuines). We can, of course, make substitutions on component expres-sions at any level, but it isnt a matter of gap filling. When we substitutebad for good in Bertie made the best choice we get Bertie made theworst choice. Wheres the gap? Freges incomplete expressions, formedby extraction, seem to be of his own creation. There are ways of build-ing well formed expression that mimic Freges operation. We build thecomplete expression by using a variable, and then adding an operator.Instead of the incomplete ((2+3_2)_), we construct x((2+3x2)x). Aswe shall see shortly, Frege is aware of such operations, but the ideologyof incomplete expressions yielding incomplete entities and completeexpressions yielding complete entities is unshakeable. So Frege cannotallow our gapless x((2+3x2)x) to stand for a function.

There is also in Freges syntactical discussions, and that of his com-mentators, more than a whiffof the problem of forming unities frompluralities, how do we obtain a single entity, a sentence, say, from a plu-

rality of words? What makes the string of words a sentence rather thanjust a list of words, as they would be (in English), if written verticallyrather than horizontally? Is it the yearning of the incomplete predicatethat is the glue that holds the parts together to form a single sentence?65

And is it the yearning of the function itself that allows the function to

65 On unsaturatedness as glue, For not all the parts of a thought can be complete; at least onemust be unsaturated, or predicative; otherwise they would not hold together. From On Conceptand Object p. 54 in Translations from Frege.



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metabolize its argument and form a value (rather than having the argu-ment just sit there, like a lump, inside the function). In reply, I wouldsay that we need to think in terms of an algebra of expressions, not theliteral pushing together of tokens. (How close together must the wordsget to form a sentence? A millimetre? An inch? A foot? A yard?)Although linguistic compounds often display their parts (for betterreadability), they dont always, as for example in better, which hasgood as a part. The claim that every language must contain a notationfor application of function to argument (or some other syncategore-matic expression that includes a notion for application of function to

argument) derives from this point of view. But every logician familiarwith Polish notation knows this to be false. (Note that juxtaposition isnot a symbol.)

In sum, (1) we dont need incomplete expressions to account for theunity of compound expressions, (2) we dont need incomplete expres-sions to account for the syntactical structure of compound expressions,(3) even if we counted the formation of compound expressions as gapfilling, we would still not need (or want) arbitraryincomplete expres-sions formed by literally extracting any part from a compound expres-sion (because not every extraction corresponds to a parsing), and (4) toform an expression for a function from an expression for one of the val-ues of the function, we proceed not by way of extraction but by way ofaddition (of variables and an operator).

Hence, there is no foundation for the theory of unsaturated func-tions in a syntactical theory of incomplete expressions.66

1.2.3 Functions objectified

According to Frege, it is these incomplete expressions that denote func-tions, which, in homage to Freges grammar, are likewise incomplete.However, each incomplete function has a corresponding course-of-val-ues, which is a complete thing, an object. In exact analogy, each incom-plete expression can be completed byfilling its gap with a variable andprefixing a variable binding operator. For example, from the incom-plete expression ((2+3_2)_) we obtain the complete x((2+3x2)x).Let us call such a complete expression a course-of-values abstract.

66 Freges distinction is, I believe, founded on an error, the error of thinking that syntax requiresthe notion of an incomplete symbol. (I argued that this is an error in my1964 Dissertation Founda-tions of Intensional Logic.) This error, I believe, may stem from a more deep-seated error, the errorof thinking that any unity with parts must have an incomplete part to lock the other parts to-gether. Regarding functions, it is in the nature of a function that yields a value when applied to anelement of its domain. If those metaphors of functional activity (as compared to the inertness ofmere correlations) make functions unsaturated, so be it.



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